Proof of a conjecture of Dahmen and Beukers on counting integral Lam\'{e} equations with finite monodromy

In this paper, we prove Dahmen and Beukers’ conjecture that the number of integral Lam'{e} equations with index $n$ modulo scalar equivalence with the monodromy group dihedral $D_N$ of order $2N$ is given by $$L_n(N)=\frac{1}{2}(\frac{n(n+1)\mathrm{\Psi }(N)}{24}-(a_n\varphi (N)+b_n\varphi (\frac{N}{2})))+\frac{2}{3}{\epsilon }_n(N)\mathrm{.}$$ Our main tool is the new pre-modular form $Z_{r,s}^{(n)}(\tau )$ of weight $n(n+1)\mathrm{/}2$ introduced by Lin and Wang \cite{LW2} and the associated modular form $M_{n,N}(\tau ):={\prod }_{(r,s)}{Z}_{r,s}^{(n)}(\tau )$ of weight $\mathrm{\Psi }(N)n(n+1)\mathrm{/}2$, where the product runs over all $N$-torsion points $(r,s)$ of exact order $N$. We show that this conjecture is equivalent to the precise formula of the vanishing order of $M_{n,N}(\tau )$ at infinity: $$v_{\mathrm{\infty }}(M_{n,N}(\tau ))=a_n\varphi (N)+b_n\varphi (N\mathrm{/}2)\mathrm{.}$$ This formula is extremely hard to prove because the explicit expression of $Z_{r,s}^{(n)}(\tau )$ is not known for general $n$. Here we succeed to prove it by using certain Painlev'{e} VI equations. Our result also indicates that this conjecture is intimately connected with the problem of counting pole numbers of algebraic solutions of certain Painlev'{e} VI equations. The main results of this paper has been announced in \cite{Lin-CDM}.